Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(26,30)(28,36)(31,33)(34,40);(1,2,5,6)(3,7,8,4)(9,11,14,17)(10,15,13,19)(12,18,20,21)(16,23,22,24)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,2)(3,7)(4,8)(5,6)(10,13)(11,17)(12,20)(23,24)(25,26,29,30)(27,33,35,31)(28,37,36,32)(34,38,40,39);(2,6)(3,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,20)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,21)(19,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,19,17,15)(12,22,20,16)(18,24,21,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gi,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968687);